in developing the Theory of Parallel Lines, 45S' 



on a formeroccasion*. M. Varignon,M.EtienneBezoutand many 

 others^ have suggested that case of the 29th proposition of EucUd, 

 which states that the exterior angle is equal to'the interior and op- 

 posite. From this, down to Mr. Stevelly's, we have had definitions 

 of all sorts. The list commences with such as fulfil Dr. Whate^ 

 ly^s idea of what a mathematical definition should be, and ter- 

 minates with such as are inadequate and useless. There is one 

 test that every logician would employ in examining the practical 

 value of ordinary definitions, which should be applied to these. 

 It depends on the relative ease with which we find the properties 

 flow from the definitions. The number of writers who have 

 gone out of their way to get a definition which involves in the 

 deductions from it, to use Dr. Lardner^s phrase, ^^ a labyrinth of 

 complicated and indirect demonstration," is remarkable. 



Eor the purpose of assisting in the choice of a definition of 



parallels, I made an analysis of the definitions of a circle and a 



square. The advantages derived from an examination of Euclid^s 



definitions are not confined to the development of the analogies 



then found to exist. Let us, for instance, take four of the most 



important : — 



I. Riffht line : — That which lies evenly between its extremities* 



II. Right angles : — When a right line standing on another right 



line makes the two adjacent angles equal. 



III. Circle : — A continued line, having a point within it, from 



which point all right lines to the continued line are 

 equal. 



IV. Parallel lines :— Such as cannot meet when produced. 



Of the first it is sufiicient to say that it is not framed in the 

 accurate language of geometry. In fact a geometer, as such, 

 can attach no meaning to the phrase lying " evenly." The de->; 

 finition has therefore never been used. The second and third 

 are clear, adequate, and perfectly logical. The fourth is certainly 

 clear enough, but it appears to be inadequate, and, as a defini- 

 tion, to be illogical. To remedy the defect of the first and 

 fourth, Euclid has given two axioms. The necessity for employ- 

 ing some positive property in treating parallel lines is thus cleslrly 

 stated in Dr. WhewelFs ' Essay on Mathematical Reasoning : '•—* 

 " We must have some positive property, besides this negative 

 one, in order to complete our reasonings respecting such lines. 

 We have, in fact, our choice of several such self-evident proper- 

 ties, any of which we may employ for our purpose, as geometers 

 well know ; but with Our naked definition, as they also know^ 

 we cannot proceed to the end." Instead^ however, of choosing' 

 some positive property of parallel lines, Euchd, unfortunately^ 

 attempted to meet the difficulty (a difficulty arising in the fii'st 

 * Phil. Mag. vol. xii. p. 372. : 



